Nowadays this result can be easily obtained by means of the ergodic theory.Borel derived his theorem from the law of large numbers. One can also derive Borel's theorem by proving first that a number \(x \in [0,1)\)  is normal in base \(b\)  if and only if the sequence \((b^n x), n \in \mathbb {N}\) , is uniformly distributed \(\bmod \, 1\)  and invoking a theorem due to Weyl (see [1]}, Theorem 4.1 on p.32), which states that for any sequence \((a_n)\)  of distinct integers, the sequence \((a_n x)\)  is uniformly distributed \(\bmod \, 1\)  for almost every \(x\) . Various equivalent forms of normality are summarized in Theorem REF  below. For historical comments on the notion of normal number and the controversies that Borel's paper generated, see Appendix in [2]}.
Indeed, consider the map \(T_b:[0,1] \rightarrow [0,1]\)  defined by \(T_b x = bx \, \bmod 1\) .
Transformation \(T_b\)  is Lebesgue measure preserving and ergodic,
and so by the pointwise ergodic theorem, we have for any \(f \in L^1 (\lambda ),\)
\(\lim _{N \rightarrow \infty } \frac{1}{N} \sum _{n=0}^{N-1} f(T_b^n(x)) = \int f \, d \lambda \quad \text{for a.e. } x.\) 
